Warner had to be physically restrained by teammates, including Paine, at tea on day four of the first Test after raging at comments allegedly made about his wife by de Kock.
Australia and South Africa have until Thursday morning AEDT to officially respond to the reports, with both having the option to appeal.
The sledging war between David Warner and Quinton de Kock has reached the next level, with the South African batsman's sister weighing into the saga via social media.
Warner was hit with a level-two charge of bringing the game into disrepute, and de Kock a level one, under the players' code of conduct. Very proud countries, both sides.
Australia calls 'Bull' on David Warner tantrum afrer Quinton de Kock fight
Lehmann said the incident had detracted from what he described as "a great game of cricket", which Australia won by 118 runs.
And #OnThisDay in 1929, David Sheppard, the first ordained minister to play Test cricket was born.
"We like to make them feel uncomfortable out there, but we don't cross the line and bring people's wives and family into it". "As long as it doesn't get personal, that's probably the key".
Match referee Jeff Crowe is believed to be looking into the incident but there has been no comment yet from the International Cricket Council.
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Former spinner Warne tweeted: "Chat, banter, sledging has always been apart of any series between SA & Oz". It's the way we play our cricket against them. "If you are saying something you've got to take it and that's the opinion of Quinton", he said.
It is alleged that De Kock made utterances about Warner's spouse, Candice, which prompted an emotional response by Warner as the players made their way into their respective changing rooms.
Australian coach Darren Lehmann expects the coaches and captains of both teams to "chat" before the second Test against South Africa starting in Port Elizabeth on Friday after a bad-tempered start to the series.
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South Africa's manager Mohammed Moosajee claims Warner peppered de Kock with personal barbs during play.
"From what I've heard there was a lot of personal stuff being said, to and from".
Australian great Adam Gilchrist also took to Twitter to voice an opinion, calling the spat "ugly".
"We play our best cricket when we're aggressive, we're in the fight together, we're hunting as a pack", Smith said.
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